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Question: Answered & Verified by Expert
If $f(x)=\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)$, then $f^{\prime}\left(\frac{\pi}{3}\right)=$
MathematicsDifferentiationJEE Main
Options:
  • A $\frac{1}{2(1+\cos x)}$
  • B $\frac{1}{2}$
  • C $\frac{1}{4}$
  • D None of these
Solution:
1564 Upvotes Verified Answer
The correct answer is: $\frac{1}{2}$
$\begin{aligned} & f(x)=\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)=\tan ^{-1}\left[\tan \frac{x}{2}\right]=\frac{x}{2} \\ & \Rightarrow f(x)=\frac{1}{2} . \text { Hence } f^{\prime}\left(\frac{\pi}{3}\right)=\frac{1}{2} .\end{aligned}$

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